D(H) -u'+qu Hu(x) = -~fix2(X)+q(x)u(x) = Au(x), H= -~x2 +q On a local uniqueness result for the inverse Sturm-Liouville problem

Ark. Mat., 39 (2001), 361-373

(~) 2001 by Institut Mittag-Leftter. All rights reserved

On a local uniqueness result for the inverse Sturm-Liouville problem

Kim Knudsen

A b s t r a c t . A new and fairly elementary proof is given of the result by B. Simon [S], t h a t the potential in a Sturm-Liouville operator is determined by the asymptotics of the associated m-function near - ~ . The proof given is based on relations between the classical transformation operators and the m-function.

1. I n t r o d u c t i o n In this paper we study the Sturm-Liouville operator

d 2

H = - ~ x 2 +q

on L2([0, co)) and the related Sturm-Liouville problem

d2u

(1)

Hu(x) = -~fix2(X)+q(x)u(x) = Au(x),

x E [0, oc),

(2) ~(0) =0.

We assume that qELI([0, cr is real valued. Under these assumptions it is well known (cf. [CL, p. 255, Problem 4]) that H is of limit-point type at infinity and selfadjoint on the domain

D(H)

-= {u E L2([0, co)) l u, u' E ACloc([0, oc)), u(0) = 0,

- u ' + q u

E L2([0, cr See [CL], [J] or [LS] for the theory of singular Sturm-Liouville problems. For a modern treatment see [Pe].

The special solution to (1) defined by the conditions (2) and u~(0)=l is called the regular solution and denoted by ~)(x, A).

362 Kim Knudsen

Since H is of limit-point type at infinity we can for A~cr(H) define u(x. A) to be the unique solution to (1) in L2([0, co)) satisfying u(0, A ) = I , the so-called Weyl solution. Associated with (1) is the m-function defined by

(3) re(A; q) = u'(0, A)

for A not an eigenvalue of H. Since the spectrum of H is real and bounded from below ([LS, Theorem 3.1]) there is a constant C > 0 such that the m-function is defined for A C C \ [ - C , c~).

The main result of this paper is a new proof of the following uniqueness result.

T h e o r e m 1.1. ([S]) Let ql,q2ELl([O, oc)) be real potentials for two Sturm- Liouville problems and let ml and m2 be the associated m-functions. Assume there is an a>O such that

m ( - k 2 ; q l ) - m ( - k 2 ; q 2 ) :o(e-ak(1-e)), as k---+~, for every r Then ql(x)=q2(x) for a.e. x e [0, 89

As a corollary to Theorem 1.1 we recover the well-known result by [Bo], [GL]

and [M] that the m-function (or equivalently the spectral measure associated with (1) and (2)) determines the potential q.

Note that since two potentials ql,q2EL~oc([O. ~c)) satisfying ql(x)=q2(x) for a.e. xE [0, 89 have associated m-functions satisfying

m(-k2;ql)-m(-k2;q2)=o(-ak(1-~-)), as k--+oc,

for all r (see [S, Theorem A.I.1]), Theorem 1.1 is valid under the less restrictive assumption that ql, q2 ELiot(J0, oc)).

In the paper IS] a new mathematical object is introduced and by this new formalism the result is proved. As observed in [GS2] this new object is closely related to a certain transformation operator relating the regular solutions to dif- ferent Sturm-Liouvitle problems, and the main idea in the present paper is to give an elementary proof of Theorem 1.1 within the framework of these transformation operators.

Recently Gesztesy and Simon [GS1] and Bennewitz [Be] have given different new proofs, which are considerably shorter than the original proof.

The outline of the paper is the following: First we review the concept of trans- formation operators and prove several estimates concerning these operators. Next we derive an equation relating the Weyl solution to a particular transformation kernel. This relation then gives a relation between the m-function and the kernel through a kind of Laplace transform. At last we derive a relation between the dif- ferent transformation kernels and by this relation and a uniqueness theorem for a related hyperbolic partial differential equation with Cauchy-data we prove Theo- rem 1.1.

O n a local uniqueness result for t h e inverse Sturm-Liouville problem 363

2. Transformation

operators

Let ql and q2 be potentials and let r and 02 be the regular solutions to the associated Sturm-Liouviiie problems. Then there exists a unique transformation kernel K independent of A such that

(4)

4)2(X,~)=4)l(X,)~)'~- j~0 x fi:(x,t)Ol(t.A)dt.

This is the Levitan-Povzner representation relating solutions to different S t u r m - Liouville problems ([Po I and [L D,

In the special case q2 =0, the regular solution is 4)2(x, A ) = s i n ( x v f A ) / v ~ , and the kernel denoted by - L satisfies

s i n ( x v ~ ) =

OI (X, ~ )-- ~0 x L(x, t )4)l (t.

A)

dt.

(5) r

Similarly, when ql =0, the kernel is denoted by K and satisfies (6) 4)2(x, A ) - sin(xv/'A) ~ . sin(tx/A )

v ~ + t K(x,t) - ~ dt.

It is easily seen by inserting (4) in (1) and making use of the initial conditions of 4)1 and 4)2 that the kernel/~ must solve the Goursat problem

(7)

~.:~(~,

^{t ) -}

~:,,(.,

t) + (q~ (t) - qa(x) ) R ( x . t) = o,

2 dF,(x,x) =q2(x)-ql(x),

~(x, 0)=0.

(x,t) ED.

x>_O.

x_>O, where D = { ( x , t ) E R 2 J 0 < t < x } .

The following lemma shows that the problem (7) is well posed.

L e i n m a 2.1.

Assume ql, q2EL l([O, oc) ). Then

(7)

has a unique solution IrE C~ Moreover, if ql,

q2eCJ([0, co)),

then fi:eCI+J(D) for

j e Z + .

In any case we have the estimate

(8)

IK( x, t)l -<

f(

^x+O/2 Iq2 (Y) - q l (Y)I

dy

J o

exp(~(x-t)/2~ (x+t)/2 ds)

x Iq2(r + s ) - q l ( r - - s ) ]

dr

364 KimKnudsen

for

0 < t < x .

The solution operator

(ql,q2)~-+K

is a continuous map in the sense, that if (q~n), q(n))__+ (ql,

q2),

as n--+ oc, in

L1 ([0, c~)) x L1 ([0, c~))

then ~[(n)(x, t)--+~[(x, t), as n--+ oc , for ( x, t ) E D , uniformly on compact subsets.

Proof.

The idea is to change coordinates and then formulate the problem as a Volterra integral equation of the second kind. This equation is then solved by iteration.

The change of variables x = ~ + r / , t = ~ - r / , defines the function

which solves

r/)=

- - , ( ~ , r/)-a(~+r/, ~-r/)k(~, r/)=0,

O~k

0 < r/<~, O(Or/

~ k ( ~ , O ) = f ( ~ ) ,

d

~>_0, k(~, ~) = o, ~ >_ o,

where

a(x, t ) = q 2 ( x ) - q l (t)

and

f ( x ) = 8 9 (x)).

Integration with respect to 7/ over the interval [0, r/] and then integration with respect to ~ over the interval Jr/, ~] yields the Volterra equation

(9)

k(~, 7/) = a(~'+r/', ~'-r/')k(~',

r/') dr~' d~' + f(u) du.

If we define the operator A on

C(D)

by

/Z

Ak(~,

71) = a(~'+r/', ~ ' - r/')k(~', r/') dr/' d~', then the equation (9) has the form

/:

(10)

( I - A ) k ( ~ , r/) = f(u) du

= F(~, r/).

Since for

cEC(D)

the inequality

(11)

[mnc(~,r/)[ ~ (

sup Ic(~',r/')[)n.!

[ a ( r + s , r - s ) [ d r d s

\o<v'_<~:'_<r

O n a local uniqueness result for t h e inverse Sturm-Liouville problem 365

can be established by induction, the operator

( I - A )

can be inverted by a convergent Neumann series. The unique solution k is thus obtained from (10). Moreover, the convergent Neumann series yields the estimate

- - \ O < _ r l ' < _ ( ' < g \ J O , I s

from which (8) follows.

The regularity of k(~, ~) and of K(x, y) is obtained from the integral equa- tion (9).

Since both the right- and left-hand side of (10) depend continuously on qE LI([0, oc)), the solution operator is continuous in the specified sense. []

Next we study the special case of the transformation kernel L. In this case the partial differential equation is given by

L , ~ ( x , t ) - L t t ( x , t ) + q ( t ) L ( x , t ) = O , (x,t) e D ,

(13)

2 d L ( x , x ) = q ( x ) , x>O,

L(x, O) = O, x >_ O.

In the following lemma we exploit (8).

L e m m a 2.2.

Let qELl([O, oo)). Then

(14)

IL(x,t)[ <

[[q[[L'

exp(x[[q[[L,),

O < t < x .

Moreover, 2Lt(2x, O)-q(x) is continuous and estimated by

(15)

[2Lt(2x,

0)-q(x)[ <_ [[q[[~l

exp(x[[q[[L,)

Iy qeCl([o, co)), then

(16)

ILt(x,t)l_<Ce

=llqllL', 0 < t < x , (17)

ILtt(x,t)l<Ce

xllqllc', 0 < t < x ,

where the constant C may depend on q.

Pro@

The problem (13) is identical to (7) with ql =q, (/2=0 and ~ ' = - L . Changing variables defines the function I(~, r/)=L((+rl, ~-r/) which because of (9) solves the equation

( i s ) l ( L v ) = -

q(~'-r/)l(~',v')dv'dg'+-~ du

366 Kim Knudsen and because of (8) is estimated by

/0' (/07/ )

(19) II(~, ~/)1 ~ 2 ]q(u)l du exp Iq(r-s)l drds .

Since for 0<7/_<~,

- - J 0 J 0 - - r /

the estimate (19) yields

/o' (/0 ~ )

II(~,n)l-

I q ( u ) l d u e x p 7? I q ( u ) l d u ,

which implies (14) for qELI([0, oc)).

Differentiating (18) yields a.e. that (20)

(21)

f n , , , 1 ~ 1

l~(~,~)=- q(~-71)l(~,o )d~l +-~q(~)=- _ q(u)l(~,~-u)du+ q(~), l,(~, ~) = q(71-v')l(y, rl') drl'- q(~'-rl)l(~', 7?) c~'- ~q(y)

// ^{fo,~ 1}

-= q(u)l(Th O-u) du- q(u)l(rl+u, 7?) du- ~q(o).

fo 2x

/o ' '

2Lt(2x, O)-q(x) = - 2 q(u)l(x,x-u)du=- q(~v)l(x,x--~v) dr,

from which the continuity and (15) follow.

When qEC~([0, c~)) we also deduce

]l~(~, Y)' <_ C exp(Tl f~ ]q(u)] du) , 'lu(~,~)] <_C exp(~ /~ ]q(u)' d u)

we find

Hence l~((, r/)- 89 and lu(~, rj)+ 89 are continuous functions. Since

Lt(x,t)=l(l~(x2t, x-t~ l fx+t

On a local uniqueness result for the inverse Sturm-Liouville problem 367 from which (16) follows. The estimate (17) follows similarly by differentiating (20) and (21) once again, since

Ltt (x, t) = -~ 2 ' '

and since q and qP are compactly supported. []

We now give a result about the Cauchy problem for the partial differential equation in (7).

L e m m a 2.3. Let A b = { ( x , t ) E R 2 i O < t < x , t + x < b } for bE[O, oc). Let ql,q2E Ll([0, b]), fEC([O,b]), geLl([O,b]). Then

B:~x(x, t ) - K u ( x , t ) + ( q l ( t ) - q 2 ( x ) ) K ( x , t) = O, (x, t) E Ab,

~'(x, 0) = f ( x ) , x E [0, b], B[t(x,O)=g(x), x E [0, b], has a unique solution h'EC(/~b).

Proof. T h e proof follows the same lines as the proof of L e m m a 2.1 (see [K] for more details). []

3. R e l a t i o n b e t w e e n t h e m - f u n c t i o n a n d a t r a n s f o r m a t i o n k e r n e l In this section we prove a relation between the Weyl solution and the transfor- mation kernel L. This result leads to the connection between L and the m-function given by

//

(22) m ( - k 2 ; q) = - k - Lt(x, O)e -xk dx.

Note that (22) can be seen as a combination of IS, equation (2.3)] and [GS2, equa- tion (9.11)]. We give a direct proof below.

We now establish a relation between u and L when

qeC~ ([0, oc)).

L e m m a 3.1. Let qEC~([0, cx~)) and let L be the transformation kernel (5).

Then for k> IIq{IL 1 ,

(23) u(t, - k 2) = e -tk - L(x, t)e -~k dx Jt

is the Weyl solution to (1).

368 Kim Knudsen

Proof.

It will be shown that uEL2([O, c~)) and that u solves (1) as well as u ( 0 , - k 2 ) = l .

According to (14),

IL(z,t)t<IIqlIL, exp(xllqllL, ),

which implies that

u ( t , - k 2)

is well defined by (23) for

k>IMIL1.

Since e-tk

E L 2 (

[0, c~)) and

f ^~ t)e -xk dx 1 e -t(k-IIqHL') E

L2([O, c~)),

L(x, <- k-IlqtlL---~

it follows that

u(t,-k2)EL2([0,

oo)).

By Lemma 2.1 the assumption qeCl([0, cc)) ensures that

LEC2(D).

Differ- entiating (23) then yields

u'(t, - k 2) = - k e -tk +L(t, t)e -tk - Lt(x, t)e -xk dx,

(24)

u " ( t , - k 2) = k 2 e - t k + d L(t,t)e-tk-kL(t,t)e-tk+Lt(t,t)e-tk -fit -~c- Ltt(x,t)e -xk dx,

since the estimates (16) and (17) justify differentiating under the integration sign.

Since L solves (13), we have (25)

- ft~

and integration by parts gives

- ~ L~(x, t)e -xk d~= L~(t, t)e-~k-~ f f L~(x, t)e-~ ex

(26) =

Lx(t, t)e -xk-kL(t, t)e - ~ k - k 2 g(x, t)e -~k dx.

Inserting (25) and (26) in (24) gives

u"(t,-k2)= ( k 2 + d L(t,t)+Lt(t,t)+Lx(t,t))e-~k-(k2+q(t)) ~t~L(x,t)e-Xk dx

= (k2+q(t))u(t,-k2),

which is (1).

Moreover, u ( 0 , - k 2 ) = l since L(x,0)=0. []

The above result is the main ingredient in the proof of the relation (22), but to obtain the result for general qELI([0, oc)) we need the following continuity result.

On a local uniqueness result for the inverse Sturm-Liouville problem 369 L e m m a 3.2. Let qELI([0, cr and suppose qnCLl([O, co)), Itq-qnllLl-+O, as n--~co. Then m(kZ;qn)-+m(k2;q), as n-+co, pointwise for every k E C with I m k sufficiently large.

Proof. For k E C with Im k sufficiently large, the Weyl solution is given by u(x, k 2) = f ( x , k)

F(k) '

where f is the Jost solution to (1) and F is the Jost function (cf. [CS]). The result follows since the map q~-~f(x, k) is continuous on LI([0, co)) for fixed (x, k). []

We are now able to prove (22).

L e m m a 3.3. For

k>l]qiiL~

the equation (22) is valid.

Proof. Assume qEC01([0, co)). From (23) we have u(t, - k 2) = e -tk + L(x, t)e -xk dx

which because of the estimate (15) gives

(27) u'(t, - k 2) = - k e -tk - L ( t , t)e-tk + Lt(x, t)e -xk dx.

The result then follows by inserting t = 0 in (27) since L(0, 0 ) = 0 and m is defined by (3).

For general qELI([0, co)) let (qn)nez+ be a sequence in C1([0, cr with IIq,~II1 <

C < k such that lim.__,~ IIq-qnlll =0. Because of (14)

/0

Lt(x,O;qn)e-Xk dx-+

/0

Lt(x,O;q)e-Xk dx, as n--+co, by dominated convergence. The result then follows from Lemma 3.2. []

4. C o n n e c t i o n b e t w e e n d i f f e r e n t t r a n s f o r m a t i o n k e r n e l s

In this section we give a result connecting the transformation kernels LI and L2 associated with two Sturm-Liouville problems and the relative transformation kernel .~.

370 Kim Knudsen

L e m m a 4.1. Assume ql, q2ELl([0, oc)). Let Li be the transformation kernel given by (5) associated with the problem

-~"(x) +q,(x)u(x) = ~u(x),

u(0) =0.

i---1, 2, and let f( be the relative transformation kernel given by (4). If ( L1)t ( x, O) = (L2)t(x,O) in LI([0, a]) for some a>0, then h't(x,0)--0 in Ll([0, a]).

Proof. The kernels L1, L2 and K satisfy (5) and (4), respectively, that is (28) sin(v~x) v/~ - r ~ ) - fro x Li(x,t)Oi(t.~)dt. i = 1,2,

L

(29) r I) = 01(x, I ) + K(x,t)Ol(t,,X)dt.

Denote by//2 the kernel associated with q2 given by (6), that is

sin(v~x) sin(v'~t)

(30) ,2(x, ~) = _{v~ ~} ff _{g2(x,t) v~ dt.}

Combining (28) for i = 1 with (30) and interchanging the order of integration yields

i;

r Ll(X,t)Ol(t,)t)dt

+ ffoXK2(x,t)(Ol(t,A)-iotLi(t,s)cbl(s.)~)ds) dt

= Ol(x,)~)+So x (K2(x,t)-Ll(x,t)-itXK2(x,S)Ll(S,t)ds)Ol(t,A) dt.

Since the kernel .~ is unique we find by (29) that

l

D 2 ( z , t ) = K 2 ( z , t ) - L l ( x , t ) - K2(x,s)Ll(s,t)ds.

Hence

St x

~[t(x,t)=(K2)t(x,t)-(L1)t(x,t)+K2(x,t)Ll(t,t)- K2(x,s)(L1)t(s,t)ds

On a local uniqueness result for the inverse Sturm -Liouville problem 371 for almost all (x, t ) E D , and since (L1)t(x,

O)=(L2)t(x,

O) in L ~ ([0. a]) and L~ (0, 0 ) = O, we get

/;

(31)

Kt(x,O)=(K2)t(x,O)-(L2)t(x,O)- K2(x,s)(L2)t(s,O)ds

for almost all x E [0, a].

On the other h a n d combining (28) for i = 2 with (30) yields

/o x

02(x, A) = r ~)--

L2(x,

t)02(t, A)

dt

+~h 89 dt,

and interchanging the order of integration gives

]ix (K2(x,t)-L2(x,t)- ft~K2(x,s)L2(s,t)ds)•

Using the fact t h a t the generalized Fourier transform is unitary yields

f"

K2(x,t)-L2(x,t)- K~(x,s)L2(s,t)ds=O,

0 < t < x . The result is now obtained by combining this equation with (31). []

5. T h e u n i q u e n e s s t h e o r e m

The last ingredient before we give the new proof of Theorem 1.1 is an inversion result for the Laplace transform,

L e m m a 5.1. ([S])

Let f c L 1

^{([0, a])}

and assume that g(z)= fo f(Y)e-ZY dy sat- isfies the relation

g(x)=o(e-ax(1-e)), as x-+~c.

for all

6>0.

Then f=~O.

We are now able to prove the main theorem.

Proof of Theorem

1.1. By Lemma 3.3

(32)

m(-k~;ql)-rn(-k2;q2) = ((L1)t(x'O)-(L2)t(x'O))e-Xk dx

=o(e-ak/1-~)), as k--+~c, for all ~>0.

372 K i m K n u d s e n

Since qeLl([0, o o ) ) a n d

12(Li)t(x,O)-q~(89 <cle c2z

by (15), we have for k >

c2 t h a t

f ~ I(L1)~(x,O)-(n2)t(x,O)le -~k dx =o(e-ak(l-~)),

^{a s}

k--+

for all c > 0 . Hence (32) gives

~oa((nl)tix, O)-(n2)t(x,O))e-~k dx=o(e-ak(1-E)), as k--+c~,

for all s >0. By Lemma 5.1 we get

(L1)t(x,O)-(L2)t(x,O) =0

for a.e. z 6 [0, el.

Lemma 4.1 now yields, that the relative transformation kernel K satisfies

~[t(x, O)=

0 for a.e. x6[0, a]. Since K is the unique solution to (7), the function K , in partic- ular, solves

I(~ix, t)-~[ttix, t)-iql(x)-q2it))~[ix, t)

^{= 0 ,}

ix, t) e D,

K ( x , 0 ) = 0 , x e [0, a], K t ( x , 0 ) = 0 , x e [0, a].

Since A~ C D this problem has according to Lemma 2.3 a unique solution K E C ( / ~ ) . Hence/~(x, t ) - 0 , ( x, t)CA~. Moreover, since K has a first order derivative almost everywhere and

~-~[(X,X)-~ql(x)--q2(x)

for a.e. x e [0, 89 0 =

we have the result. V7

Acknowledgement.

The ideas described in this paper are a part of the author's master thesis (June 1999) supervised by Associate Professor Erik Skibsted, De- partment of Mathematical Sciences, Aarhus University. I thank Erik for fruitful discussions during spring 1999 as well as useful comments on this paper. I also thank Professor Arne Jensen at Aalborg University for comments on the paper.

R e f e r e n c e s

[Be] BENNEWITZ,

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y"+(A-q(x))y=O,

ⁱⁿ

Den 11te Skandinaviske Matematikerkongress ( Trondheim, 1949),

pp. 276- 287, Johan Grundt Tanums Forlag, Oslo, 1952.

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Received April 18, 2000

in revised form December 14, 2000

Kim Knudsen

Department of Mathematical Sciences Aalborg University

Fredrik Bajers Vej 7G DK-9220 Aalborg O Denmark

email: kim@math.auc.dk